A relation $ R $ is defined on $ \mathbb{Z} $ by $ xRy $ if $ x \cdot y \geq 0 $. Prove or disprove the following:
(a) $ R $ is reflexive
(b) $ R $ is symmetric
(c) $ R $ is transitive
(a) If $ xRx $ then $ x \cdot x \geq 0 $ for all $ x $ in $ \mathbb{Z} $. This is true because $ (-a)(-a) = a $, for all $ a $ in $ \mathbb{Z} $.
(b) If $ xRy $ then we want $ yRx $ for all $ x, y $ in $ \mathbb{Z} $. This is true because $ ab = ba $ for all $ a, b $ in $ \mathbb{Z} $.
(c) Now I am stuck because I know that "If $ xRy $ and $ yRz $ we want that $ xRz $. This is true because …" but how do I say that multiplication is transitive.
Best Answer
HINT: Is $-1\,R\,0$? Is $0\,R\,1$? Is ... ?